Stochastic Analysis and Particle Filtering of the Volatility

Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 103)

Abstract

Let \(S = {({S}_{t})}_{t\in {\rm{IR}}_{+}}\) be an \({\rm{IR}}_{+}\)-valued semimartingale based on a filtered probability space \((\Omega,\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\rm{IR}}_{+}}, \rm{IP})\) which is assumed to be continuous. The process S is interpreted to model the price of a stock. A basic problem arising in Mathematical Finance is to estimate the price volatility, i.e., the square of the parameter σ in the following stochastic differential equation
$$d{S}_{t} = \mu {S}_{t}\,dt + \sigma {S}_{t}\,d{W}_{t}$$
where \(W = {({W}_{t})}_{t\in {\rm{IR}}_{+}}\) is a Wiener process. It turns out that the assumption of a constant volatility does not hold in practice. Even to the most casual observer of the market, it should be clear that volatility is a random function of time which we denote σ t 2. Itô’s formula for the return \({y}_{t} =\log ({S}_{t}/{S}_{0})\) yields
$$d{y}_{t} = \left (\mu -\frac{{\sigma }_{t}^{2}} {2} \right )\,dt + {\sigma }_{t}\,d{W}_{t}\quad {y}_{0} = 0$$
(11.1)
The main objective is to estimate in discrete real-time one particular sample path of the volatility process using one observed sample path of the return. As regards the drift μ, it is constant but unknown. Under the so-called risk-neutral measure, the drift is a riskless rate which is well known; actually one finds that μ does not cancel out, for instance, when calculating conditional expectations in a filtering problem. For this argument no change of measure is required, we work directly in the original measure \(\rm{IP}\), and μ has to be estimated from the observed sample path of the return as well.

Keywords

Covariance Volatility 

References

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Signal Processing and Electronic SystemsÉcole Supérieure d’ÉlectricitéGif sur YvetteFrance

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